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Navigation & Position formula solvers

Categories — 73 of them, 1787 solvers in all
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Compass to True Heading (Variation and Deviation)

T=C+D+VT = C + D + V

Navigation & PositionThe TVMDC chain in one line: a compass heading corrected by deviation for the ship's own iron and by variation for the earth's field, giving the true heading a chart is drawn in. East corrections are positive and are added going this way.

Cross Track Error

ext=Rarcsin⁡(sin⁡d13R sin⁡(θ13−θ12))e_{xt} = R\arcsin\left(\sin\frac{d_{13}}{R}\,\sin\left(\theta_{13}-\theta_{12}\right)\right)

Navigation & PositionThe perpendicular distance from the intended track to where you actually are, on a sphere: the number a plotter shows on its deviation bar. Positive is right of track, negative is left.

Dead Reckoning Position

Δφ=S tcos⁡CR\Delta\varphi = \frac{S\,t\cos C}{R}

Navigation & PositionThe classic DR advance: speed times time gives the distance run, and the cosine of the course splits it into the change of latitude. The oldest position-finding method there is, and still the one that catches an instrument failure.

Destination Latitude from Course and Distance

φ2=arcsin⁡(sin⁡φ1cos⁡dR+cos⁡φ1sin⁡dRcos⁡θ)\varphi_2 = \arcsin\left(\sin\varphi_1\cos\frac{d}{R} + \cos\varphi_1\sin\frac{d}{R}\cos\theta\right)

Navigation & PositionWhere a great circle course of a given length puts you: the latitude reached from a starting latitude, an initial true bearing and a distance run. The answer carries the change of longitude with it.

Distance from Pace Count

d=n Ld = n\,L

Navigation & PositionDistance covered from a pace count times your calibrated pace length. The compass gives the direction; this is how a navigator measures the how-far.

Distance to the Visible Horizon

D=h(2R+h)D = \sqrt{h\left(2R + h\right)}

Navigation & PositionHow far away the sea horizon is from a given height of eye: the tangent line from the observer to the sphere. The geometric figure, before atmospheric refraction pushes it out by about a tenth.

Estimated Time En Route

t=dVgt = \frac{d}{V_g}

Navigation & PositionDistance divided by ground speed. Trivial arithmetic, and the single most common place a navigation plan goes wrong, because the speed put into it is so often the airspeed or the log speed rather than the speed actually made good.

Great Circle Distance (Haversine)

d=2Rarcsin⁡sin⁡2φ2−φ12+cos⁡φ1cos⁡φ2sin⁡2λ2−λ12d = 2R\arcsin\sqrt{\sin^{2}\frac{\varphi_2-\varphi_1}{2} + \cos\varphi_1\cos\varphi_2\sin^{2}\frac{\lambda_2-\lambda_1}{2}}

Navigation & PositionThe shortest distance between two points on a sphere, from their latitudes and longitudes. Sinnott's haversine form, which keeps its precision on short legs where the older law-of-cosines version loses it.

Ground Distance on a Slope

g=m2+r2g = \sqrt{m^{2} + r^{2}}

Navigation & PositionThe distance your legs travel on a slope, from map distance and rise. Pythagoras, and forgiving: a 20 % grade adds only 2 % to the distance.

Height by Clinometer

H=dtan⁡θ+eH = d\tan\theta + e

Navigation & PositionHeight of a tree or cliff from a paced distance and a clinometer angle, plus your eye height — the sight line starts at your eye, not your boots.

Initial Great Circle Bearing

θ=atan2⁡ ⁣(sin⁡Δλ cos⁡φ2,  cos⁡φ1sin⁡φ2−sin⁡φ1cos⁡φ2cos⁡Δλ)\theta = \operatorname{atan2}\!\left(\sin\Delta\lambda\,\cos\varphi_2,\; \cos\varphi_1\sin\varphi_2 - \sin\varphi_1\cos\varphi_2\cos\Delta\lambda\right)

Navigation & PositionThe course at the moment of departure along the great circle to a destination, measured clockwise from true north. It is the starting bearing only — on any track but a meridian or the equator it changes the whole way.

Longitude to Solar Time Difference

Δt=4 Δλ\Delta t = 4\,\Delta\lambda

Navigation & PositionSolar time difference from a longitude difference: 4 minutes per degree, because the Earth turns 360° in 24 hours. Sun time, not zone time.

Map Scale to Real Distance

d=m Sd = m\,S

Navigation & PositionReal distance from a map measurement times the scale denominator. On a 1:50,000 sheet every centimetre stands for 50,000 cm of world — half a kilometre.

Naismith's Rule (Hiking Time)

t=d5 km/h+h600 m/ht = \frac{d}{5\,\text{km/h}} + \frac{h}{600\,\text{m/h}}

Navigation & PositionNaismith's 1892 walking-time estimate: an hour for every 5 km on the map plus an hour for every 600 m of climb. A planning floor, not a promise.

Rhumb Line Distance

d=R(φ2−φ1)2+q2(λ2−λ1)2d = R\sqrt{\left(\varphi_2-\varphi_1\right)^{2} + q^{2}\left(\lambda_2-\lambda_1\right)^{2}}

Navigation & PositionThe length of the track that crosses every meridian at the same angle: the route you actually steer when you set one course and hold it. Longer than the great circle, and the stretching factor q comes from the Mercator latitude.

Set and Drift of the Current

Dr=Δn2+Δe2tD_r = \frac{\sqrt{\Delta n^{2} + \Delta e^{2}}}{t}

Navigation & PositionThe current the water was making, worked backwards from the gap between where dead reckoning said you were and where the fix put you. Drift is its speed; set is the true direction it flows toward.

Wind Correction Angle

WCA=arcsin⁡(Wsin⁡θV)\mathrm{WCA} = \arcsin\left(\frac{W\sin\theta}{V}\right)

Navigation & PositionHow far off the desired track to point the nose so the wind carries you along it instead of off it. Only the crosswind component matters: the along-track part of the wind changes the speed, never the heading.

Wind Triangle Ground Speed

Vg=V2−W2sin⁡2θ−Wcos⁡θV_g = \sqrt{V^{2} - W^{2}\sin^{2}\theta} - W\cos\theta

Navigation & PositionWhat the aircraft is actually making over the ground once the wind has had its say: true airspeed, wind speed, and the angle between the wind and the intended track. The E6B's whole reason for existing.